Single annulus LP estimates for Hilbert transforms along vector fields

被引：23

作者：

Bateman, Michael ^{[1
,2
]}

机构：

[1] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90095 USA

[2] Univ Cambridge, Dept Pure Math & Math Stat, Ctr Math Sci, Cambridge CB3 0WB, England

来源：

REVISTA MATEMATICA IBEROAMERICANA | 2013年 / 29卷 / 03期

关键词：

Carleson's theorem; time-frequency analysis; Stein's conjecture; Zygmund's conjecture; differentiation of vector fields; Hilbert transform in direction of a vector field; MAXIMAL OPERATORS; AVERAGES; SETS;

D O I：

10.4171/RMI/748

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We prove L-P estimates, p is an element of (1, infinity), on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when p > 2 were proved by Lacey and Li. This paper also contains key technical ingredients for a companion paper with Christoph Thiele in which L-P estimates are established for the full Hilbert transform. The operators considered here are singular integral variants of maximal operators arising in the study of planar differentiation problems.

引用

页码：1021 / 1069

页数：49

共 12 条

[1]

BATEMAN M., ANAL PDE IN PRESS

[2] MAXIMAL AVERAGES ALONG A PLANAR VECTOR FIELD DEPENDING ON ONE VARIABLE [J].

Bateman, Michael .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2013, 365 (08) :4063-4079

[3]

Bateman M, 2009, P AM MATH SOC, V137, P955

[4] KAKEYA SETS AND DIRECTIONAL MAXIMAL OPERATORS IN THE PLANE [J].

Bateman, Michael .

DUKE MATHEMATICAL JOURNAL, 2009, 147 (01) :55-77

[5] Maximal operators over arbitrary sets of directions [J].

Katz, NH .

DUKE MATHEMATICAL JOURNAL, 1999, 97 (01) :67-79

[6]

Lacey M, 2000, MATH RES LETT, V7, P361

[7]

Lacey M., 2010, Mem. Am. Math. Soc., V205

[8] Maximal theorems for the directional Hilbert transform on the plane [J].

Lacey, Michael T. ;

Li, Xiaochun .

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2006, 358 (09) :4099-4117

[9] DIFFERENTIATION IN LACUNARY DIRECTIONS [J].

NAGEL, A ;

STEIN, EM ;

WAINGER, S .

PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 1978, 75 (03) :1060-1062

[10]

SJOGREN P, 1981, ANN I FOURIER, V31, P157

← 1 2 →